《半经典微局部分析 Semiclassical Microlocal Analysis》课程教学资源（英文讲义）Lec11 THEORY OF SEMICLASSICAL PSDOS：BOUNDEDNESS

LECTURE11:L?-THEORY OF SEMICLASSICALPsDOs:BOUNDEDNESSIn the previous several lectures, we have studied the definition and basic prop-erties of semiclassical pseudodifferential operators, but mainly as an operator actingon(Rn).However, as wehave seen, in quantum part (=the spectral part) of thestory the natural space should be a Hilbert space:(Rn) is not.In the next sev-erallecturesweshall studyproperties of semiclassicalpseudodifferentialoperatorsas linear operators acting on L?(Rn), or in cases we need more regularity, acting onthe Sobolev spaces H"(R").1.L?-BOUNDEDNESS OFOP(a)FOR SCHWARTZSYMBOLSSuppose a = a(r, ) E J(R2n) is a Schwartz function. Then as we have seen,the operator aw, or more generally, the operator Opi(a) for any t e [0,1], maps'(Rn) into(R"). In particular, these operators are linear maps from L?(R")intoL?(IRn).It turns out thatfor a Schwartz symbol, the operator Opt(a)is alwaysa bounded linear operator (and as we will prove next time, is a compact operator)on L?(R"). In what follows we will provide two different proofs of this fact.I Schur's test.To prove the L2-boundedness of linear operators like Opt(a) which are definedby Schwartz kernels, a very useful criterion is the following Schur's test:Lemma1.1 (Schur's Test).Let K :RnxRn→Cbe a continuous function satisfying[K(r, y)]dy< +o0Ci = supandC2= supIK(r, y)]dr < +00,Rand let A be the linear operator with Schwartz kernel K:Au(r) =K(r,y)u(y)dyThenAisa boundedlinear operatorfromL?(Rn)toL?(Rn)with(1)IIAllc(L2(R") ≤(CIC2)Proof. For any u E L?(IR") the Cauchy-Schwartz inequality gives[Au(r)?≤ / [K(a,g)]dy./ [K(r,9)]lu(g)P’dy ≤C1 /[K(r, )]lu(g)'dy

LECTURE 11: L 2 -THEORY OF SEMICLASSICAL PSDOS: BOUNDEDNESS In the previous several lectures, we have studied the definition and basic prop￾erties of semiclassical pseudodifferential operators, but mainly as an operator acting on S (R n ). However, as we have seen, in quantum part (=the spectral part) of the story the natural space should be a Hilbert space: S (R n ) is not. In the next sev￾eral lectures we shall study properties of semiclassical pseudodifferential operators as linear operators acting on L 2 (R n ), or in cases we need more regularity, acting on the Sobolev spaces Hs (R n ). 1. L 2 -boundedness of Opt ~ (a) for Schwartz symbols Suppose a = a(x, ξ) ∈ S (R 2n ) is a Schwartz function. Then as we have seen, the operator ba W , or more generally, the operator Opt ~ (a) for any t ∈ [0, 1], maps S 0 (R n ) into S (R n ). In particular, these operators are linear maps from L 2 (R n ) into L 2 (R n ). It turns out that for a Schwartz symbol, the operator Opt ~ (a) is always a bounded linear operator (and as we will prove next time, is a compact operator) on L 2 (R n ). In what follows we will provide two different proofs of this fact. ¶ Schur’s test. To prove the L 2 -boundedness of linear operators like Opt ~ (a) which are defined by Schwartz kernels, a very useful criterion is the following Schur’s test: Lemma 1.1 (Schur’s Test). Let K : R n×R n → C be a continuous function satisfying C1 = sup x Z Rn |K(x, y)|dy < +∞ and C2 = sup y Z Rn |K(x, y)|dx < +∞, and let A be the linear operator with Schwartz kernel K: Au(x) = Z Rn K(x, y)u(y)dy. Then A is a bounded linear operator from L 2 (R n ) to L 2 (R n ) with (1) kAkL(L2(Rn)) ≤ (C1C2) 1 2 Proof. For any u ∈ L 2 (R n ) the Cauchy-Schwartz inequality gives |Au(x)| 2 ≤ Z Rn |K(x, y)|dy · Z Rn |K(x, y)||u(y)| 2 dy ≤ C1 Z Rn |K(x, y)||u(y)| 2 dy. 1

2LECTURE 11:L?-THEORY OF SEMICLASSICAL PSDOS:BOUNDEDNESSIntegrating withrespectto ,weget(C1- / K(r,y)llu(g)Pdy) dr≤CiCall/2IIAul/≤/口L?-boundedness for PsDOs with Schwartz symbolsAs an immediate consequence,Theorem 1.2. If a = a(r, E) is a Schwartz function, then for any t e [0, 1],Opt(a) : L?(Rn) → L?(Rn)is a bounded linear operator withI(a)lc(L2(R")≤sup sup, (ga)(,)/(R)lal<≤n+1Proof. Recall that the Schwartz kernel of the operator Opt(a) iset(a-)-Sa(tr +(1 -t)y,s)d =b(tr+(1 -t)yka(r,y)(2元h)n(2元h)*节where b is the“partial Fourier transform" of a given bye-it"a(r,s)deb(r, z) = Fe-→2[a(r,E)] =Since a is a Schwartz function, b is also a Schwartz function (Check this!).Thus1[6(tr + (1 - t)y, "二-[kg(r,y)]dr =/da(2元h)n方1[b(y - thz,z)[dz(2元)n1e)-n-1dz . sup Kz)n+1b(y,z2)l(2元)n.2e1(2)-n-1dz·sup,sup, l[Fe2al](y,z)l~(Rn)≤(2元）ny[al<n)-n-1dz·supsup1[Fg-(oga)](y,z)/L(Rn)(2元)nyJal<n1(z)-n-1dz·sup sup Il(oga)(y,E)llzi(R)≤Ci :(2元)nyJal<n+and similarly1(2)-n-dz·sup sup I(oga)(r,E)li(R)[ka(r, y)]dy ≤ C2(2元)nJa<n+口Now the conclusion follows from Schur's test

2 LECTURE 11: L 2 -THEORY OF SEMICLASSICAL PSDOS: BOUNDEDNESS Integrating with respect to x, we get kAuk 2 L2 ≤ Z Rn  C1 · Z Rn |K(x, y)||u(y)| 2 dy dx ≤ C1C2kuk 2 L2 . ¶ L 2 -boundedness for PsDOs with Schwartz symbols. As an immediate consequence, Theorem 1.2. If a = a(x, ξ) is a Schwartz function, then for any t ∈ [0, 1], Opt ~ (a) : L 2 (R n ) → L 2 (R n ) is a bounded linear operator with kOpt ~ (a)kL(L2(Rn)) ≤ sup x sup |α|≤n+1 k(∂ α ξ a)(x, ξ)kL1(Rn ξ ) . Proof. Recall that the Schwartz kernel of the operator Opt ~ (a) is k a t (x, y) = 1 (2π~) n Z Rn e i ~ (x−y)·ξ a(tx + (1 − t)y, ξ)dξ = 1 (2π~) n b(tx + (1 − t)y, y − x ~ ), where b is the “partial Fourier transform” of a given by b(x, z) = Fξ→z[a(x, ξ)] = Z Rn e −iξ·z a(x, ξ)dξ. Since a is a Schwartz function, b is also a Schwartz function (Check this!). Thus Z Rn |k a t (x, y)|dx = 1 (2π~) n Z Rn |b(tx + (1 − t)y, y − x ~ )|dx = 1 (2π) n Z Rn |b(y − t~z, z)|dz ≤ 1 (2π) n Z Rn hzi −n−1 dz · sup y,z∈Rn |hzi n+1b(y, z)| ≤ 1 (2π) n Z Rn hzi −n−1 dz · sup y sup |α|≤n+1 kz α [Fξ→za](y, z)kL∞(Rn z ) ≤ 1 (2π) n Z Rn hzi −n−1 dz · sup y sup |α|≤n+1 k[Fξ→z(∂ α ξ a)](y, z)kL∞(Rn z ) ≤ C1 := 1 (2π) n Z Rn hzi −n−1 dz · sup y sup |α|≤n+1 k(∂ α ξ a)(y, ξ)kL1(Rn ξ ) and similarly Z Rn |k a t (x, y)|dy ≤ C2 := 1 (2π) n Z Rn hzi −n−1 dz · sup x sup |α|≤n+1 k(∂ α ξ a)(x, ξ)kL1(Rn ξ ) . Now the conclusion follows from Schur’s test.

3LECTURE11:L?-THEORYOFSEMICLASSICALPSDOS:BOUNDEDNESSRemark. One can also prove the L?-boundedness of Opt(a) directly as follows: Westart with Weyl's decomposition (c.f.the formula (11)in Lecture 9,page 7)[op(e(r+)] [(Fr)ss)-(w)a(y, )dydn.(Opt(a))(r) =(2元h)2nSince the operator Opi(et(u+n-s) is unitary on L2(R"), the triangle inequality im-pliesI[(Fn)(s,s)(y,n)al(y, n)]dydn =IIOp;(a)l(L2(R") ≤(2元h)2n(2m)2~1/ (g.5)(0,)al/ 1.It remains to estimate F(s.t)-(yn)alli. Note that here we are using the usualFourier transform, not the semiclassical one. So our estimate is uniform w.r.t. h.We state and prove a general result:The L1-norm of the Fourier transform of a Schwartz function can becontrolled by the Ll-norm of its partial derivatives:Lemma 1.3. There erists a constant C = Cn such that for anyaE(Rn),Falli≤Cnsuplaallzi.al<n+1Proof.We haveIIFal/μ = / [Fa()] ds ≤ [ (5)-n-1 d I(s)n+1 Fa()lL≤Cn,supllsaFallL≤Cnsuploallzi.[al<n+1al≤n+1As a consequence, we get:Proposition 1.4. For any a E (R2n), and any t e [0, 1](2)IIOp(a)lc(L2(R")≤Cnsup IIoallLI.lall<2n+2.L?BOUNDEDNESSFORMOREGENERALSYMBOLSBoundedness of symbols v.s. L?-boundedness of operators.We would like to extend the L?-boundedness result we proved above to semi-classical pseudo-differential operators with symbols in more general classes. This isnot always possible. For example,·NeithertheoperatorQj=“multiplication by

LECTURE 11: L 2 -THEORY OF SEMICLASSICAL PSDOS: BOUNDEDNESS 3 Remark. One can also prove the L 2 -boundedness of Opt ~ (a) directly as follows: We start with Weyl’s decomposition (c.f. the formula (11) in Lecture 9, page 7) (Opt ~ (a))(x) = 1 (2π~) 2n Z R2n h Opt ~ (e i ~ (y·x+η·ξ) ) i [(F~)(s,ξ)→(y,η)a](y, η)dydη. Since the operator Opt ~ (e i ~ (y·x+η·ξ) is unitary on L 2 (R n ), the triangle inequality im￾plies kOpt ~ (a)kL(L2(Rn)) ≤ 1 (2π~) 2n Z R2n |[(F~)(s,ξ)→(y,η)a](y, η)|dydη = 1 (2π) 2n kF(s,ξ)→(y,η)akL1 . It remains to estimate kF(s,ξ)→(y,η)akL1 . Note that here we are using the usual Fourier transform, not the semiclassical one. So our estimate is uniform w.r.t. ~. We state and prove a general result: The L 1 -norm of the Fourier transform of a Schwartz function can be controlled by the L 1 -norm of its partial derivatives: Lemma 1.3. There exists a constant C = Cn such that for any a ∈ S (R n ), kFakL1 ≤ Cn sup kαk≤n+1 k∂ α akL1 . Proof. We have kFakL1 = Z Rn |Fa(ξ)| dξ ≤ Z Rn hξi −n−1 dξ · khξi n+1Fa(ξ)kL∞ ≤ Cn sup |α|≤n+1 kξ αFakL∞ ≤ Cn sup kαk≤n+1 k∂ α akL1 . As a consequence, we get: Proposition 1.4. For any a ∈ S (R 2n ), and any t ∈ [0, 1], (2) kOpt ~ (a)kL(L2(Rn)) ≤ Cn sup kαk≤2n+1 k∂ α x,ξakL1 . 2. L 2 boundedness for more general symbols ¶ Boundedness of symbols v.s. L 2 -boundedness of operators. We would like to extend the L 2 -boundedness result we proved above to semi￾classical pseudo-differential operators with symbols in more general classes. This is not always possible. For example, • Neither the operator Qj = “multiplication by xj”

4LECTURE11:L?-THEORYOFSEMICLASSICALPSDOS:BOUNDEDNESSnor the operatorhaPj=ioriisbounded onL?(Rn)(check thisstatementbyproviding counterexamples!).This is reasonable since neither the function i (the classical counterpartof Qi) nor the function E, (the classical counterpart of P) are boundedfunctions..On the otherhand,ifa(r)is a bounded continuous function, namely a(r)l≤Cfor all r eRn, then obviously the operatorOpi(a) = Ma(a) =“multiplication by a(r)"is abounded operatoron L?since/ la(r)u(r)’drI/Ma(a) ul 2 =≤Cllulz2.. Similarly if a(s) is a bounded function, i.e. [a($)I ≤ C, thenOp(a) = Fr-1 0 Ma(s) 0 Fhis a bounded operator on L?, since the Plancherel's Theorem (c.f. Lecture4, Prop. 1.7)implies1CI/Oph(a)ullL2 =(2h)/l/Ful/2 = Il/2.(2 h)/lMa()Fu/2 Calderon-Vaillancourt Theorem: Idea of proof.So one may guess that for any bounded symbol a(r, E), the operator Oph(a) isa bounded operator on L?. Unfortunately this is not quite true in general. (I needan example here.) However, we will prove that if a E S(1), namely if we assume thesymbol a itself togetherwith all its derivatives are bounded, then Opt(a)is boundedon L?(Rn):Theorem 2.1 (Calderon-Vaillancourt). If a E S(1), then the operatorOpt(a) : L?(Rn) → L?(Rn)is a bounded linear operator on L?(Rn)with(3)IIOpi(a)lc(L2(R) ≤C Z hl/2 sup|8°alR21lal<Mnfor some universal constant MlIn the rest of this lecture, we prove Calderon-Vaillancourt's theorem.The ideais the following:IWe can take Mn to be 10n + 6

4 LECTURE 11: L 2 -THEORY OF SEMICLASSICAL PSDOS: BOUNDEDNESS nor the operator Pj = ~ i ∂ ∂xj is bounded on L 2 (R n ) (check this statement by providing counterexamples!). This is reasonable since neither the function xj (the classical counterpart of Qj ) nor the function ξj (the classical counterpart of Pj ) are bounded functions. • On the other hand, if a(x) is a bounded continuous function, namely |a(x)| ≤ C for all x ∈ R n , then obviously the operator Opt ~ (a) = Ma(x) = “multiplication by a(x)” is a bounded operator on L 2 since kMa(x)ukL2 = Z |a(x)u(x)| 2 dx−1/2 ≤ CkukL2 . • Similarly if a(ξ) is a bounded function, i.e. |a(ξ)| ≤ C, then Opt ~ (a) = F −1 ~ ◦ Ma(ξ) ◦ F~ is a bounded operator on L 2 , since the Plancherel’s Theorem (c.f. Lecture 4, Prop. 1.7) implies kOpt ~ (a)ukL2 = 1 (2π~) n/2 kMa(ξ)F~ukL2 ≤ C (2π~) n/2 kF~ukL2 = CkukL2 . ¶ Calderon-Vaillancourt Theorem: Idea of proof. So one may guess that for any bounded symbol a(x, ξ), the operator Opt ~ (a) is a bounded operator on L 2 . Unfortunately this is not quite true in general. (I need an example here.) However, we will prove that if a ∈ S(1), namely if we assume the symbol a itself together with all its derivatives are bounded, then Opt ~ (a) is bounded on L 2 (R n ): Theorem 2.1 (Calderon-Vaillancourt). If a ∈ S(1), then the operator Opt ~ (a) : L 2 (R n ) → L 2 (R n ) is a bounded linear operator on L 2 (R n ) with (3) kOpt ~ (a)kL(L2(Rn)) ≤ C X |α|≤Mn ~ |α|/2 sup R2n |∂ α a| for some universal constant M. 1 In the rest of this lecture, we prove Calderon-Vaillancourt’s theorem. The idea is the following: 1We can take Mn to be 10n + 6

LECTURE11:L?-THEORYOFSEMICLASSICALPSDOS:BOUNDEDNESS5.Wefirst decompose a into countably many compactly-supported symbolsa=Em am.This can be done by choosing any partition of unity 1 = m Xmsuch that each Xm is compactly supported, and letting am = Xma. ThenformallywehaveOpk(a) =Opk(am),and by compactness of supp(am), each Opi(am) is L?-bounded.? In general, if A=Am, to conclude the boundedness of A from the bound-edness of Am's,a necessary condition weneed is thatthe bound of Am's is uniform for all m.In view of (2),this“uniformlyboundedness"can be fulfilled if we chooseour partition of unity in a“"uniform" way.-The“uniformly boundedness" of components is still not enough, sincethere may be “interactions" between different Am's. Of course the bestdream is that if we can choose these Am's so that there are“no interac-tion",or in other words,if these Am's are"orthogonal"to each other.(namelyifthedecompositionA=mAmisan“"orthogonaldecompo-sition" A =@mAm), then the“uniformly boundedness"of Am's doesimply the boundedness of A. But that's only a dream: we can't chooseourdecompositionmOpt(am)tobeorthogonal.-However, the dream shed a light on the correct direction! The“orthogonality"or the“no-interaction condition"implies that Am o Am=Oform' + m. This is too strong, since it is enough to assume "almost-orthogonality", or in other words, it is enough if we have a nicely-controlled“interaction". Back to our decomposition. Although we can't make Opt(am)oOpi(am) = 0forall m'+m,wecanmakeOpt(am)oOpt(am)=O(h)formostm+m.In fact,according to Corollary 1.4 inLecture9,wehaveOpt(am) 0 Oph(am) = O()if supp(am)nsupp(am)= 0. In other words, wedo have“almost-orthogonality"of Opt(am),if we start with"almost-disjoint"symbols am's.[In other words"almost-orthogonality"is the quantum analogue of the“"almost disjointnessoffunctionsl.In summary, here is how we prove thetheorem:A We first carefully choose a partition of unity 1 = Em Xm, in a uniformway,so that the resulting decomposition a =(xma)decompose a into asummation of almost disjoint compactly-supported symbols am = Xma.B We then control the interaction between different Opi(am)s

LECTURE 11: L 2 -THEORY OF SEMICLASSICAL PSDOS: BOUNDEDNESS 5 • We first decompose a into countably many compactly-supported symbols a = P m am. This can be done by choosing any partition of unity 1 = P m χm such that each χm is compactly supported, and letting am = χma. Then formally we have Opt ~ (a) = X m Opt ~ (am), and by compactness of supp(am), each Opt ~ (am) is L 2 -bounded. • In general, if A = PAm, to conclude the boundedness of A from the bound￾edness of Am’s, – a necessary condition we need is that the bound of Am’s is uniform for all m. In view of (2), this “uniformly boundedness” can be fulfilled if we choose our partition of unity in a “uniform” way. – The “uniformly boundedness” of components is still not enough, since there may be “interactions” between different Am’s. Of course the best dream is that if we can choose these Am’s so that there are “no interac￾tion”, or in other words, if these Am’s are “orthogonal” to each other, (namely if the decomposition A = P m Am is an “orthogonal decompo￾sition” A = L m Am), then the “uniformly boundedness” of Am’s does imply the boundedness of A. But that’s only a dream: we can’t choose our decomposition P m Opt ~ (am) to be orthogonal. – However, the dream shed a light on the correct direction! The “orthog￾onality” or the “no-interaction condition” implies that Am ◦Am0 = 0 for m0 6= m. This is too strong, since it is enough to assume “almost￾orthogonality”, or in other words, it is enough if we have a nicely￾controlled “interaction”. • Back to our decomposition. Although we can’t make Opt ~ (am)◦Opt ~ (am0) = 0 for all m0 6= m, we can make Opt ~ (am) ◦Opt ~ (am0) = O(~ ∞) for most m0 6= m. In fact, according to Corollary 1.4 in Lecture 9, we have Opt ~ (am) ◦ Opt ~ (am0) = O(~ ∞) if supp(am)∩supp(am0) = ∅. In other words, we do have “almost-orthogonality” of Opt ~ (am), if we start with “almost-disjoint” symbols am’s.[In other words, “almost-orthogonality” is the quantum analogue of the “almost disjointness” of functions]. In summary, here is how we prove the theorem: A We first carefully choose a partition of unity 1 = P m χm, in a uniform way, so that the resulting decomposition a = P(χma) decompose a into a summation of almost disjoint compactly-supported symbols am = χma. B We then control the interaction between different Opt ~ (am)s